Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Claim. For a finite covering system with pairwise distinct odd moduli greater than one, the least common multiple of the moduli has at least six distinct prime factors. This is the corollary of equations (12)--(15) of M. A. Berger, A. Felzenbaum and A. S. Fraenkel, Necessary condition for the existence of an incongruent covering system with odd moduli II, Acta Arith. 48 (1987), no. 1, 73--79, compiled on the library's six-prime corollary page. The paper's main theorem is a polynomial necessary condition on the prime powers in the period, proved by a forest correction to the union bound over prime-adic boxes; the corollary evaluates it at the five smallest odd primes, where the worst case gives , and so excludes five primes. It improves the five primes of the authors' Part I (Acta Arith. 45 (1986), no. 4, 375--379).
Covers. The case of Problem 7 whose moduli involve at most five distinct primes: no such distinct odd covering exists, so every distinct odd covering has period at least . The unrestricted question stays open.
Depends on. Nothing in this wiki; the theorem is the paper's own.
Acceptance. Refereed: Acta Arithmetica 48 (1987), no. 1, 73--79, doi:10.4064/aa-48-1-73-79. Not reviewed: the site labels the problem VERIFIABLE, an open label, and its commentary does not cite the paper. Not formalized: no Lean proof of the corollary is recorded; the library's compilation is author-recorded coverage by this project and awards nothing here.